Mathematics is not a spectator sport: successful students of mathematics grapple with ideas for themselves. *Distilling Ideas* presents a carefully designed sequence of exercises and theorem statements that challenge students to create proofs and concepts. As students meet these challenges, they discover strategies of proofs and strategies of thinking beyond mathematics. In order words, *Distilling Ideas* helps its users to develop the skills, attitudes, and habits of mind of a mathematician and to enjoy the process of distilling and exploring ideas.

*Distilling Ideas* is an ideal textbook for a first proof-based course. The text engages the range of students' preferences and aesthetics through a corresponding variety of interesting mathematical content from graphs, groups, and epsilon-delta calculus. Each topic is accessible to users without a background in abstract mathematics because the concepts arise from asking questions about everyday experience. All the common proof structures emerge as natural solutions to authentic needs. *Distilling Ideas* or any subset of its chapters is an ideal resource either for an organized Inquiry Based Learning course or for individual study.

A student response to *Distilling Ideas*: "I feel that I have grown more as a mathematician in this class than in all the other classes I've ever taken throughout my academic life."

### Table of Contents

1. Introduction

2. Graphs

3. Groups

4. Calculus

5. Conclusion

Annotated Index

List of Symbols

Abouth the Authors

### Excerpt: Ch. 3.12 The Man Behind the Curtain (p. 85)

Many people mistakenly believe that mathematics is arbitrary and magical, or at least that there is some secret knowledge that math teachers have but won't share with their students. Mathematics is no more magical that the Great and Powerful Wizard of Oz, who was just behind a curtain. The development of mathematics is directed by a few simple principles and a strong sense of aesthetics. To develop the ideas of group theory we followed a path of guided discovery. Let's look back on the journey and let the guiding strategies emerge from behind the curtain.

Group theory has many applications, ranging from internet credit card security and cracking the Enigma Code to solving the Rubik's Cube and generalizing the quadratic formula. Our introduction to group theory has only exposed us to a tiny portion of the theorems living in the wild.

We started with familiar activities: adding, multiplying, telling time, and putting blocks in a matching hole. The definition of a group emerged by distilling the essential features and commonalities from these specific examples in the process that we call abstraction. These examples generalized to produce cyclic groups, the dihedral groups, and symmetric groups in the process we call exploration. We defined generators, subgroups, products, and homomorphisms to help us discover and describe the wide variety of groups that we caught in our definition-net. We saw patterns that led us to conjectures about the size of subgroups; we justified these insights, producing theorems. We applied those insights to extend our exploration of the concept of groups.

This investigation represents a complete cycle of inquiry, and it especially emphasizes and deepens the skill of definition exploration. We explored a mathematical idea in its own right, hoping to describe the rich mosaic of possibilities that an abstract diefinitiion could capture. In particular, trying to *classify* all groups was one of the motivations that guided our explorations. Let us reflect on these new tools in our inquiry tool belt...

Preview: p. 117

### About the Authors

**Brian Katz** is an Assistant Professor of Mathematics at Augustana College in Rock Island, Illinois. He received his BA from Williams College in 2003 with majors in mathematics, music and chemistry and his PhD from The University of Texas at Austin in 2011, concentrating in algebraic geometry. While at UT Austin, Brian received the Frank Gerth III Graduate Excellence Award and the Frank Gerth III Graduate Teaching Excellence Award from the Department of Mathematics. Brian is a Project NExT Fellow, supported by Harry Lucas, Jr. and the Educational Advancement Foundation.

Brian is a member of the Inquiry Based Learning community, which focuses on helping students to ask and explore mathematical questions for themselves. Brian has given many talks about mathematics, teaching, and technology at national conferences and workshops, including as a plenary speaker for the 2012 MAA IBL PREP workshop. In particular, Brian has written and spoken about ways to use student-written wikis to extend the power of IBL beyond the classroom. Brian has helped to organize the R.L. Moore Legacy Conference hosted by the Academy for Inquiry Based Learning, and serves as a mentor for new practitioners within the organization.

Brian is liberally educated and is passionate about engaging in the Liberal Arts. He sees mathematics at the core of this educational perspective: connecting to the deductive reasoning of philosophy, the structure of communication of linguistics, the abstract beauty of the arts, and all forms of critical writing and speaking. Outside of mathematics, Brian has offered an interdisciplinary first-year course called Mind and Meaning, and one of his mathematics courses has been designated as part of the core curriculum at Augustana College as a "Perspective on Human Values and Existence."

As an educator, Brian enjoys helping all students become clearer thinkers and communicators and empowering people to move from consuming to producing knowledge.

**Michael Starbird** is Professor of Mathematics and a University Distinguished Teaching Professor at The University of Texas at Austin. He received his BA degree from Ponoma College and his PhD in mathematics from the University of Wisconsin, Madison. He has been on the faculty of the Department of Mathematics of The University of Texas at Austin except for leaves including as a Visiting Member of the Institute for Advanced Study in Princeton, New Jersey, and as a member of the technical staff at the Jet Propulsion Laboratory in Pasadena, California. He served as Associate Dean in the College of Natural Sciences at UT from 1989 to 1997.

Starbird is a member of the Academy of Distinguished Teachers at The University of Texas of Austin and is an inaugural member of The University of Texas System Academy of Distinguished Teachers. He has won many teaching awards, including the 2007 Mathematical Association of America Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics; a Minnie Stevens Piper Professorship, which is awarded each year to 10 professors from any subject at any college or university in the state of Texas; the inaugural award of the Dad's Association Centennial Teaching Fellowship; the Excellence Award from the Eyes of Texas, twice; the President's Associates Teaching Excellence Award; the Jean Holloway Award for Teaching Excellence, which is the oldest teaching award at UT and is presented to one professor each year; the Chad Oliver Plan II Teaching Award, which is student-selected and awarded each year to one professor in the Plan II liberal arts honors program; and the Friar Society Centennial Teaching Fellowship, which is awarded to one professor at UT annually. He is an inaugural year Fellow of the AMS. In 1989, Professor Starbird was the UT Recreational Sports Super Racquets Champion.

Starbird's mathematical research is in the field of topology. He has served as a member-at-large of the Council of the American Mathematical Society and on the national education committees of both the American Mathematical Society and the Mathematical Association of America. He currently serves on the MAA's CUPM Committee and on its Steering Committee for the next CUPM Curriculum Guide. He directs UT's Inquiry Based Learning Project. He has given more than 200 invited lectures at the colleges and universities throughout the country and more than 20 minicourses and workshops to mathematics teachers.

Starbird strives to present higher-level mathematics authentically to students and the general public and to teach thinking strategies that go beyond mathematics as well. With those goals in mind, he wrote, with co-author Edward B. Burger, *The Heart of Mathematics: An invitation to effective thinking* (now in its 4th edition), which won a 2001 Robert W. Hamilton Book Award. Burger and Starbird have also written a book that brings intriguing mathematical ideas to the public, entitled *Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas*, published by W.W. Norton, 2005, and translated into eight foreign languages. In 2012, Burger and Starbird published *The 5 Elements of Effective Thinking*, which describes practical strategies for creating innovation and insight.

Starbird has produced five courses for The Teaching Company in their Great Courses Series: *Change and Motion: Calculus Made Clear *(1st edition, 2001, and 2nd edition, 2007)*; Meaning from Data: Statistics Made Clear, *2005*; What are the Chances? Probability Made Clear, *2007*; Mathematics from the Visual World, *2009; and, with collaborator Edward Burger, *The Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas, *2003*.*These courses bring an authentic understanding of significant ideas in mathematics to many people who are not necessarily mathematically oriented. Starbird loves to see real people find the intrigue and fascination that mathematical thinking can bring.

David Marshall, Edward Odell, and Michael Starbird wrote *Number Theory Through Inquiry*, which appeared in 2007 in the MAA's Textbook Series. *Number Theory Through Inquiry* and the new Katz-Starbird book *Distilling Ideas: An Introduction to Mathematical Thinking* are books in the newly created MAA Textbook Subseries called "Mathematics Through Inquiry." This subseries contains materials that foster an Inquiry Based Learning strategy of instruction that encourages students to discover and develop mathematical ideas on their own.